Basics of MATLAB and Beyond

The top trace in the top plot is the noisy data, and the bottom trace isthe original pure sinusoid. The lower plot clearly shows the frequency at100 Hz.Two GUI-based FFT demos can be accessed by typing demo at theprompt. Select the “Signal Processing” option, then choose the “DiscreteFourier Transform” or the “Continuous Fourier Transform”.Exercise 5 Extend the ideas in the previous example to twodimensions, as would be the case, for example, if you made measurementsin space and time, rather than time alone. Generatea two-dimensional sinusoid and explore its FFT. (Answer onpage 185.)18 Power SpectrumThe power spectrum (or power spectral density, or PSD) is a measureof the power contained within frequency intervals. The problem is thatwe only have a finite set of samples of the true signal so we can neverhave perfect knowledge about its power spectrum. A common way toestimate a PSD is to use the square of the FFT of the samples. Thesquare of the FFT is called the periodogram. The workhorse of matlab’speriodogram-based spectral estimation is the spectrum function(in the Signal Processing Toolbox). We illustrate using data similar tothe previous example of a noisy sinusoid, but we take more samples. APSD estimate can be found by typing:dt = 1/1000;t = dt:dt:8192*dt;sine = sin(2*pi*100*t);y = sine + randn(size(t));clfspectrum(y)The frequency scale is normalised to the Nyquist frequency. The middleline is the PSD estimate and the two dashed lines are the 95% confidenceintervals. Typing help spectrum reveals that there are manyparameters that you can adjust when calculating the power spectrum.matlab’s spectrum function uses the Welch method of PSD estimation, 6which divides a long signal into a number of smaller blocks, calculates6 See Alan V. Oppenheim and Ronald W. Schafer, Digital Signal Processing,Prentice-Hall, 1975, p. 553. An excellent general treatment of PSD estimation isalso given in William Press, Brian Flannery, Saul Teukolsky and William Vetterling,Numerical Recipes, Cambridge University Press, 1989.c○ 2000 by CRC Press LLC